PLANETARY BIRKHOFF NORMAL FORMS

Birkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown that the classical Poincaré variables and the RPS–variables (introduced in [6]), after a trivial lift, lead to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincaré variables) is degenerate at all orders (answering a question of M. Herman). Non-degenerate Birkhoff normal forms for partially and totally reduced cases are provided and an application to long–time stability of secular action variables (eccentricities and inclinations) is discussed.


Introduction
Let us consider the planetary (1 + n)-body problem, i.e. , the motions of 1 + n point-masses, interacting only through gravity, with one body ("the Sun") having a much larger mass than the other ones ("the planets"). A fundamental feature of this Hamiltonian system (for negative decoupled energies) is the separation between fast degrees of freedom, roughly describing the relative distances of the planets, and the slow (or "secular") degrees of freedom, describing the relative inclinations and eccentricities (of the osculating Keplerian ellipses). A second remarkable feature of the planetary system is that the secular Hamiltonian has (in suitable "Cartesian variables") an elliptic equilibrium around zero inclinations and eccentricities. Birkhoff normal form (hereafter "BNF") theory 1 comes, therefore, naturally in. Such theory yields, in particular, information on the secular frequencies (first order Birkhoff invariants) and on the "torsion" (or "twist") of the secular variables (the determinant of the second order Birkhoff invariants). Indeed, secular Birkhoff invariants are intimately related to the existence of maximal and lower dimensional KAM tori 2 , or, as we will show below ( § 6), one can infer long-time stability for the "secular actions" (essentially, eccentricities and mutual inclinations). A natural question is therefore the construction of BNFs for the secular planetary Hamiltonian. Already Arnold in 1963 realized that this is not a straightforward task in view of secular resonances, i.e. , rational relations among the first order Birkhoff invariants holding identically on the phase space. Incidentally, Arnold was aware of the socalled rotational resonance (the vanishing of one of the "vertical" first order Bikhoff invariants) but did not realize the presence of a second resonance of order 2n − 1 discovered by M. Herman (compare [10] and [1]). These resonances, apart from being an obstacle for the construction of BNFs, constituted also a problem for the application of KAM theory. This problem was overcome, in full generality, only in 2004 [10] using a weaker KAM theory involving only information on the first order Birkhoff invariants, waving the check of Kolmogorov's non-degeneracy (related to full torsion 3 ); for a short description of the main ideas involved, see [6,Remark 11.1,(iii)].
In particular the question of the torsion of the secular Hamiltonian remained open. M. Herman investigated such question thoroughly using Poincaré variables [11] but declared not to know if some of the second order Birkhoff invariants was zero even in the n = 2 case (compare the Remark towards the end of p. 24 in [11]). A different point of view is taken up in [6], where a new set of variables, called rps ("Regularized Planetary Symplectic") variables, is introduced in order to study the symplectic structure of the phase space of the planetary system. Such variables are based on Deprit's action-angles variables ( [8], [5]), which may be used for a symplectic reduction lowering by one the number of degrees of freedom. A further reduction is possible (at the expense of introducing a new singularity) leading to a totally reduced phase space, compare [6, §9] and § 5.1 below. On the reduced phase spaces, one can construct BNFs ([6, Sect 7 and 9]; § 2, § 5.1 below). Following such strategy one can show that the matrix of second order Birkhoff invariants (for the reduced system) is non-degenerate and prove full torsion. In particular, it is then possible to construct a large measure set of maximal non-degenerate KAM tori ( [6, §11]).
In this paper we consider and clarify various aspects of BNFs for the planetary system. In particular we analyze the connection between the BNF in the classical setting (Poincaré variables) and in the new setting of [6]. It turns out that after lifting in a trivial way the rps variables to the full dimensional phase space, such variables and the Poincaré variables are related in a very simple way, namely, through a symplectic map which leaves the action variables Λ (conjugated to the mean anomalies) fixed and so that the correspondence between the respective Cartesian variables is close to the identity map (and independent of the fast angles); compare Theorem 3.1 below. Since, up to such class of symplectic maps, the BNF is unique, one sees that the BNF in Poincaré variables is degenerate at all orders, answering negatively the question of M. Herman; see Theorem 2.1 below. We mention also that the construction of BNF for rotational invariant Hamiltonian (such as the secular planetary Hamiltonian) is simpler than the standard construction: in fact, one needs to assume non-resonance of the first order Birkhoff invariant for those Tay-lor modes k = 0 such that i k i = 0 (and not just k = 0); compare Appendix A. By this remark one sees that the secular resonances (both the rotational and the Herman resonance) do not really affect the construction of BNFs. In § 5.1 we discuss the construction, up to any order, of the BNFs in the totally reduced setting (generalizing Proposition 10.1 in [6]) and, for completeness, we consider ( § 5.2) the planar planetary problem (in which case the Poincaré and the rps variables coincide) and, after introducing a (total) symplectic reduction, we discuss BNFs in such reduced setting, comparing, in particular, with the detailed analysis in [11]. Finally, in § 6, we use the results of § 5.1 in order to prove that, in suitable open non-resonant phase space regions of relatively large Liouville measure, the eccentricities and mutual inclinations remain small and close to their initial values for times which are proportional to any prefixed inverse power of the distance from the equilibrium point (zero inclinations and zero eccentricities): such result is somewhat complementary to Nehorošev's original result [13], where exponential stability of the semi major axes was estabilished, but no information on possible large (order one) variation of the secular action was given.

Planetary BNF
After the symplectic reduction of the linear momentum, the (1 + n)-body problem with masses m 0 , µm 1 , · · · , µm n (0 < µ 1) is governed by the 3n-degrees of freedom Hamiltonian where x (i) represent the difference between the position of the i th planet and the position of the Sun, y (i) are the associated symplectic momenta rescaled by µ, x · y = 1≤i≤3 x i y i and |x| := (x · x) 1/2 denote, respectively, the standard inner product in R 3 and the Euclidean norm; The phase space is the "collisionless" domain of R 3n × R 3n (y, x) = (y (1) , . . . , y (n) ), (x (1) , . . . , x (n) ) s.t. 0 = x (i) = x (j) , ∀ i = j , (2.3) endowed with the standard form When µ = 0, the Hamiltonian (2.1) is integrable: its unperturbed limiting value h plt is the sum of the Hamiltonians corresponding to uncoupled Two-Body Newtonian interactions. In Poincaré coordinates -which will be reviewed in the next section -the Hamiltonian (2.1) takes the form where (Λ, λ) ∈ R n × T n ; the "Kepler" unperturbed term h k , coming from h plt in (2.1), becomes Because of rotation (with respect the k (3) -axis) and reflection (with respect to the coordinate planes) invariance of the Hamiltonian (2.1), the perturbation f p in (2.5) satisfies well known symmetry relations called d'Alembert rules, see (3.26)-(3.31) below. By such symmetries, in particular, the averaged perturbation is even around the origin z = 0 and its expansion in powers of z has the form 4 where Q h , Q v are suitable quadratic forms. The explicit expression of such quadratic forms can be found, e.g. , in [10, (36), (37)] (revised version). By such expansion, the (secular) origin z = 0 is an elliptic equilibrium for f av p and corresponds to co-planar and co-circular motions. It is therefore natural to put (2.8) into BNF in a small neighborhood of the secular origin; see, e.g. , [12] for general information on BNFs and Appendix A for Birkhoff theory for rotational invariant Hamiltonian systems. As a preliminary step, one can diagonalize (2.8), i.e. , find a symplectic transfor-mationΦ (the domainM 6n p will be specified in (2.15) below) defined by Λ → Λ and In this way, (2.8) takes the form with the average overλ off av given bỹ (2.12) The 2n real vector Ω := (σ, ς) = (σ 1 , · · · , σ n , ς 1 , · · · , ς n ) is formed by the eigenvalues of the matrices Q h and Q v in (2.8) and are called the first order Birkhoff invariants. It turns out that such invariants satisfy identically the following two secular resonances Such resonances strongly violate the usual non-degeneracy assumptions needed for the direct construction of BNFs. The first resonance, discovered by M. Herman, is still quite mysterious (see, however, [1]), while the second resonance is related to the existence of two noncommuting integrals, given by the horizontal components C 1 and C 2 of the total angular momentum C := n i=1 x (i) × y (i) of the system (compare [2]). Actually, the effect of rotation invariance is deeper: the vanishing of the eigenvalue ς n is just "the first order" of a "rotational" proper degeneracy, as explained in the following theorem, which will be proved in § 4. Let w := (u, v) = (u 1 , · · · , u 2n , v 1 , · · · , v 2n ),w := (u 1 , · · · , u 2n−1 , v 1 , · · · , v 2n−1 ) and (2.14) G , one can construct a symplectic map ("Birkhoff transformation"), with the following properties. The pull-back of the Hamiltonian (2.11) takes the form where the average f av b (Λ, w) := T n f b dl is in BNF of order s: , (2.17) P s being homogeneous polynomial in r of order s, parameterized by Λ. Such normal form is unique up to symplectic transformations Φ which leave the Λ's fixed and with thez-projection independent of l and close to the identity in w, i.e. , Furthermore, the normal form (2.16)-(2.17) is "infinitely degenerate", in the sense that H b does not depend on (u 2n , v 2n ). In particular, there exists a unique polyno-mialP s : R 2n−1 → R (parameterized by Λ) such that P s (r) =P s (r) wherer := (r 1 , · · · , r 2n−1 ) .
Remark 2.1 (i) Notice that the w-projection of M 6n b corresponds to a neighborhood of w = 0, which is small only in the 4n − 2 components of w, while it is large (maximal) in the remaining 2 components (compare Appendix B for the natural radius 2 √ G in the variables (u 2n , q 2n )). Indeed, to construct the normal form, by rotation invariance, it is not necessary to assume that all inclinations are small, but one can take the mutual inclinations to be small. This corresponds to consider 2n − 1 secular degrees of freedom (roughly, corresponding to n couples of eccentricities-perihelia and n − 1 couples of inclinations-nodes) instead of 2n. The overall inclination-node of the system (corresponding to the remaining 2 secular variables) is allowed to vary globally. (ii) Theorem 2.1 depends strongly upon the rotational invariance of the Hamiltonian (2.1), that is, on the fact that such Hamiltonian commutes with the three components of the angular momentum C. To exploit explicitly such invariance, we shall use a set of symplectic variables ("rps variables"), introduced in [6] (in order to describe the symplectic structure of the planetary N-body problem and to check KAM non-degeneracies). (iii) The rps variables are obtained as a symplectic regularization of a set of actionangle variables, introduced by Deprit in 1983 ( [8], [5]), which generalize to an arbitrary number n of planets the classical Jacobi's reduction of the nodes (n = 2). The remarkable property of the Deprit's variables is that there appear a conjugate couple (C 3 and ζ below) plus an action variable G which are integrals. Thus, the conjugate integrals are also cyclic and are responsible for the proper degeneracy of the planetary Hamiltonian. Furthermore, the rps variables have a cyclic couple ((p n , q n ) below), which foliates the phase space into symplectic leaves (the sets M 6n−2 (p n ,q n ) in (3.14) below), on which the planetary Hamiltonian keeps the same form. So, the construction of the "non degenerate part" of the normal form can be made up to any order (and is the same) on each leaf [6]. In particular, the even order of the remainder in (2.17) is due to invariance by rotations around the C-axis of the system. Finally, we prove that such normal form can be uniquely lifted to the degenerate normal form (2.17)-(2.19) on the phase space M 6n p in (2.9). The proof is based on the remarkable link between rps and Poincaré variables, described in the following section (see Theorem 3.1).

Poincaré and RPS variables
In this section we first recall the definitions of the Poincaré and rps variables 5 and then discuss how they are related. Recall that the Poincaré variables have been introduced to regularize around zero eccentricities and inclinations the Delaunay action-angle variables. Analogously, the rps variables have been introduced to regularize around zero eccentricities and inclinations the Deprit action-angle variables.
• Fix 2n positive "mass parameters 6 " M i ,m i and consider the two-body Hamil- plt as in (2.4). Assume that h i (y (i) , x (i) ) < 0 so that the Hamiltonian flow φ t h i (y (i) , x (i) ) evolves on a Keplerian ellipse E i and assume that the eccentricity e i ∈ (0, 1). Let a i , P i denote, respectively, the semi major axis and the perihelion of E i . Let C (i) denote the i th angular momentum C (i) := x (i) × y (i) .
-To define Deprit variables, consider the "partial angular momenta" (notice that C is the total angular momentum of the system) and define the "Deprit nodes" For u, v ∈ R 3 lying in the plane orthogonal to a vector w, let α w (u, v) denote the positively oriented angle (mod 2π) between u and v (orientation follows the "right hand rule").
Notice that: • Delaunay's variables are defined on an open set of full measure P 6n Del of the Cartesian phase space P 6n := R 3n × R 3n * , namely, on the set where e i ∈ (0, 1) and the nodesν i in (3.1) are well defined. • On P 6n Del and P 6n Dep , the "Delaunay inclinations" i i and the "Deprit inclinations" ι i , defined through the relations (3.6) are well defined and we choose the branch of cos −1 so that i i , ι i ∈ (0, π). Finally: • The Poincaré variables are given by (Λ, λ, z) := (Λ, λ, η, ξ, p, q), with the Λ's as in (3.4) and The rps variables are given by (Λ, λ, z) := (Λ, λ, η, ξ, p, q) with (again) the Λ's as in (3.4) and Remark 3.1 From the definitions (3.8)-(3.9) it follows that the variables are defined only in terms of the integral C. Thus, they are integrals (hence, cyclic) in Hamiltonian systems which commute with the three components of the angular momentum C (or, equivalently, in systems which are invariant by rotations).
Let φ p and φ rps denote the maps φ p : (y, x) → (Λ, λ, z) , φ rps : (y, x) → (Λ, λ, z) . The main point of this procedure is that: • The map φ p can be extended to an analytic symplectic diffeomorphism on the set P 6n Del which is defined as P 6n Del , but with e i and i i allowed to be zero.
• The map φ rps can be extended to an analytic symplectic diffeomorphism on the set P 6n Dep which is defined as P 6n Dep , but with e i and ι i allowed to be zero. The image sets M 6n max,p := φ p (P 6n Del ) and M 6n max,rps := φ rps (P 6n Dep ) are defined by elementary inequalities following from the definitions (3.7) and (3.8) (details in Appendix B). Notice in particular that • e i = 0 corresponds to the Poincaré coordinates η i = 0 = ξ i and the rps coordinates η i = 0 = ξ i ; • i i = 0 corresponds to the Poincaré coordinates p i = 0 = q i ; • ι i = 0 corresponds to the the rps coordinates p i = 0 = q i . In particular p n = 0 = q n corresponds to the angular momentum C being parallel to the k (3) -axis.
(3.12) (roughly,z are related to eccentricities-perihelia, and mutual inclinationsnodes of the instantaneous ellipses E i ). Then, M 6n max,rps can be written as is just the length of the total angular momentum expressed in rps variables as given in (2.14) and M 6n−2 max is a given subset of R n + ×T n ×R 4n−2 (compare the end of Appendix B).
• We have already observed that for rotation invariant systems the variables (p n , q n ) are cyclic. In this case, the phase space M 6n max,rps is foliated into symplectic leaves max,rps : p n = p n , q n = q n } . (3.14) In the next section, for the application to the planetary problem, we shall substitute the set M 6n−2 max in the definition (3.13) of M 6n max,rps with a smaller set M 6n−2 : compare (4.2) below.
Consider the common domain of the maps φ p and φ rps in (3.11), i.e. the set P 6n Del ∩ P 6n Dep . In particular, on such set, 0 ≤ e i < 1, 0 ≤ i i < π, 0 ≤ ι i < π. On the φ rps -image of such domain consider the symplectic map where ϕ(Λ, 0) = 0 and, for any fixed Λ, the map Z(Λ, ·) is 1:1, symplectic 7 and its projections verify, for a suitable To prove Theorem 3.1, we need some information on the analytical expressions of the maps φ p and φ rps .
• The analytical expression of the Cartesian coordinates y (i) and x (i) in terms of the Poincaré variables (3.7) is classical: pl is the planar Poincaré map. Explicitly, -The planar Poincaré map is given by 8 is the unique solution of the (regularized) Kepler equation -The Poincaré rotation matrix is given by • The formulae of the Cartesian variables in terms of the rps variables, differ from the formulae of the Poincaré map (3.18) just for the rotation matrix. Namely, one has (3.23) 8 Compare, e.g., [3].
pl is the planar Poincaré map defined above. The expression of the rps rotation matrices R (i) rps is a product of matrices Notice that the only matrix in (3.24) depending on (p n , q n ) is R * n .
Extending results proven in [6], we now show that φ rps p in (3.15) "preserves rotations and reflections"(Lemma 3.1 below). Consider the following symplectic transformations where, denoting the imaginary unit by i, Such transformations correspond, in Cartesian coordinates, to, respectively, reflection with respect to the plane x 1 = x 2 , the plane x 3 = 0 and a positive rotation of g around the k (3) -axis: where R 3 (g) denotes the matrix For future use, consider also the following transformations, which are obtained obtained by suitably combining R 1↔2 and R g : In particular, by D'Alembert rules, the expansion (2.8) follows.
Proof It is enough to prove Lemma 3.1 for the transformations in (3.26) and (3.27). But this follows from the fact that both in Poincaré variables and in rps variables the transformations in (3.28) have the form in (3.26)-(3.27).
Proof of Theorem 3.1 For the proof of (3.16) (since φ rps p is a regular map), we can restrict to the open dense set where none of the eccentricities e i or of the nodes ν i+1 orν i vanishes. In such set the angles γ i , g i , θ i and ψ i are well defined. By the definitions of λ i in (3.7) and of λ i in (3.8), one has The (3.4) and (3.5)), as well as the angles θ i and ψ j depend only on the angular momenta C (1) , · · · , C (n) ; hence, they do not depend upon λ.
With similar arguments one proves the second equation in (3.16). Injectivity of Z(Λ, ·) follows from the definitions. That, for any fixed Λ, Z(Λ, ·) is symplectic, is a general property of any map of this form which is the projection over z of a symplectic transformation (Λ, λ, z) → (Λ, λ, z) which leaves Λ unchanged. Notice now that φ rps p preserves the quantities and the quantities Therefore, it also preserves From the previous equalities one has that φ rps p sends injectively (η i , ξ i ) = 0 to (η i , ξ i ) = 0 and (p, q) = 0 to (p, q) = 0. From the analytical expressions of φ p and φ rps there follows that, when (p, q) = 0, the Poincaré variables (η, ξ) and λ and the Deprit's (η, ξ) and λ respectively coincide. Therefore, from (3.16) and (3.33), we have ϕ(Λ, 0) = 0 and the first two equations in (3.17) follow. The fact that the remainder is O(|z| 3 ) is because Z(Λ, ·) is odd in z, as we shall now check. In fact, using Lemma 3.1 with R = R − 1 or R = R − 2 , one finds that the (η, q)-projection of Z(Λ, ·) is odd in (η, q), even in (ξ, p); the (ξ, p)-projection of Z is odd in (ξ, p), even in (η, q). In particular, Z(Λ, ·) is odd in z. Equation (3.34) and the fact that Z is odd imply that (p, q) = R(p, q) + O(|z| 3 ), with R ∈ SO(2n). Since p is odd in (ξ, p) and q is odd in (η, q), one has that R is block diagonal:

Proof of the normal form theorem
For the proof of Theorem 2.1, we need some results from [6], to which we refer for details. Let H rps denote the planetary Hamiltonian expressed in rps variables: where H plt is as in (2.1) and φ rps as in (3.11). Notice that, as H plt is rotation invariant, the variables p n , q n in (3.10) are cyclic for H rps . Hence, the perturbation function f rps depends only on the remaining variables (Λ, λ,z), wherez is as in (3.12).
To avoid collisions, consider the ("partially reduced") variables in a subset of the maximal set M 6n−2 max in (3.13) of the form where A is a set of well separated semi major axes where a 1 , · · · , a n , a 1 , · · · , a n , are positive numbers verifying a j < a j < a j+1 for any 1 ≤ j ≤ n, a n+1 := ∞; B 4n−2 is a small (4n − 2)-dimensional ball around the "secular origin"z = 0. As in the Poincaré setting, the Hamiltonian H rps enjoys D'Alembert rules (namely, the symmetries in (3.27) and in (3.31)). Indeed, since the map φ rps p in (3.15) commutes with any transformations R as in (3.26)-(3.31) and H p is R-invariant, one has thatH rps is R-invariant: This implies that the averaged perturbation f av rps also enjoys D'Alembert rules and thus has an expansion analogue to (2.8), but independent of (p n , q n ): with Q h of order n andQ v of order (n − 1). Notice that the matrix Q h in (4.5) is the same as in (2.8), since, when p = (p, p n ) = 0 and q = (q, q n ) = 0, Poincaré and rps variables coincide. The first step is to construct a normal form defined on a suitable lower dimensional domain The existence of such normal form for the Hamiltonian (4.5) at any order s defined over a set of the form (4.6) is a corollary of [6, §7]. Indeed (by [6]), one can first conjugate H rps = h k + µf rps to a Hamiltoniañ so that the averagef av rps has the quadratic part into diagonal form: wherez = (η,ξ,p,q) and σ i ,ς i denote 9 the eigenvalues of the matrices Q h andQ v in (4.5). Here,φ denotes the "symplectic diagonalization" which lets Λ → Λ and where U h ∈ SO(n) andŪ v ∈ SO(n − 1) put Q h andQ v into diagonal form and will be chosen later. Notice thatφ leaves the set M 6n−2 in (4.2) unchanged. Next, we can use Birkhoff theory for rotation invariant Hamiltonians, which allows to construct BNF for rotation invariant Hamiltonian for which there are no resonance (up to a certain prefixed order) for those Taylor indices k such that k i = 0 (rather than k = 0 as in standard Birkhoff theory; compare Appendix A below). Indeed, as shown in [6,Proposition 7.2], the first order Birkhoff invariants Ω = (σ,ς) ∈ R n × R n−1 do not satisfy any resonance (up to any prefixed order s) over a (s-dependent) set A chosen as in (4.3), other than n i=1 σ i + n−1 i=1ς i = 0 andς n = 0. Thus, one can find a Birkhoff normalizationφ defined on the set (4.6), which conjugatesH rps = h k + µf rps tȏ wheref av rps is in the form (2.17), with r of dimension n + (n − 1) = 2n − 1 and Ω = (σ,ς) replacing Ω andP s as in (2.19). It is a remarkable fact, proved in [6], that both the transformationsφ andφ above leave G(Λ,z) in (2.14) unchanged (i.e., they commute with R g ). Therefore, if we denote where M 6n−2 andM 6n−2 are as in (4.2) and (4.6), respectively, we have thatφ andφ can be lifted to symplectic transformations through the identity map on (p n , q n ). Moreover: whereH rps andH rps are as in (4.7) and in (4.10), respectively; (ii)Φ rps is given by (4.9), with (p, p n ), (q, q n ), (p, p n ), (q, q n ), replacingp,q,p,q,Ū v , respectively; (iii)Φ rps is of the form (2.18) (with w andz replaced by (z, p n , q n ) and (z, p n , q n ), respectively), since a similar property holds forφ.
Proof of Theorem 2.1. We prove only existence of the normal form; uniqueness follows from the same argument of standard BNF theory: compare [12]. LetH p as in (2.11), whereΦ p is as in (2.9)-(2.10), for suitable fixed matrices ρ h , Analogously, letΦ rps ,Φ rps as in (4.14), φ rps p as in (3.15). Consider the transformation where f av b =f av rps has just the claimed form. To conclude, we have to check (2.18). It is sufficient to prove such equality (with w replaced by (z, p n , q n )) for the transformation Φ b in (4.18) (by item (iii) above). But this is an immediate consequence of (2.10), (3.17), (4.17), (4.19) and item (ii) above.
Remark 4.1 As a byproduct of the previous proof, we find that the matrices Q v in (2.8) and Q v = diag [Q v , 0] in (4.16) have the same eigenvalues, so that the invariants ς i andς i in (2.8) and (4.8) coincide (for i ≤ n − 1).

Further reductions and BNFs
In this section we discuss complete symplectic reduction by rotations, together with the respective BNFs, both in the spatial and planar cases (indeed, as in the three-body case, the planar case cannot be simply deduced from the spatial one in view of singularities). The BNFs constructed in the spatial case ( § 5.1) is at the basis of the dynamical application given in § 6.

The totally reduced planar case
Let us now restrict to the planar setting, that is, when the coordinates y (i) , x (i) in (2.1) are taken in R 2 instead of R 3 . Also in this case, in view of the presence of the 1 , a (total) symplectic reduction is available (compare, also, [9]). In the case of the planar problem, the instantaneous ellipses E i defined in § 3 become coplanar and both the Poincaré variables (Λ, λ, z) and rps variables (Λ, λ, z) reduce to the planar Poincaré variables. Analytically, the planar Poincaré variables can be derived from (3.7) by setting θ i = 0 and disregarding the p and q. To avoid introducing too many symbols, we keep denoting the planar Poincaré variable where A can be taken as in (4.3) above and B 2n the (2n)-dimensional open ball around the origin, whose radius (related to eccentricities, as in the spatial case), is chosen so small to avoid collisions; beware that z = (η, ξ), here, is 2n-dimensional. The planetary Hamiltonian in such variables is given by H pl (Λ, λ, z) = h Kep (Λ) + µf pl (Λ, λ, z) obtained from H p in (2.5) by putting, simply, p = 0 = q; clearly, also the expression of the averaged perturbation, f av pl , can be derived in the same way from (2.7). Since, in particular, the "horizontal" first order Birkhoff invariants σ do not satisfy resonances of any finite order s on 11 A, the Birkhoff-normalization up to the any order can be constructed in the planar case and it coincides with the expression of f av b in (2.17), where one has to take w = (u, v) =: (η,p), (ξ,q) = (η, 0), (ξ, 0) . We recall in fact that the transformation Φ b in Theorem 2.1 sends injectively p = 0 =q to p = 0 = q and hence the restriction Φ b |p =0=q performs the desired normalization in the planar case. Let us denote by the planar Birkhoff-normalized system, that is, the system such that the averaged perturbationf av (Λ,z) is in BNF: the BNF of order 4 is given by The asymptotic evaluation of the first order invariants σ and especially of planar torsionτ in (5.11) for general n ≥ 2 can be found in the paper by J. Féjoz [10] and in the notes by M. Herman [11]. However, since the asymptotics considered in such papers is slightly than the one considered in [6] for the general spatial case different 12 , we collect here the asymptotic expressions of σ andτ as they follow from [6] (compare also below for a short proof): • The first order Birkhoff invariants σ into (5.11) satisfy • The second order Birkhoff invariantsτ into (5.11) satisfy, for 13 n = 2, and for 14 n ≥ 3, where δ := a −3 n (5.14) withτ of rank (n − 1) and • Eq. (5.12) implies in particular non resonance of the σ j 's into a domain of the form of (4.3) (with a j , a j depending on s). 12 In [10] , [11] the semi major axes a 1 < · · · < a n are taken well spaced in the following sense: at each step, namely, when a new planet (labeled by "1") is added to the previous (n − 1) (labeled from 2 to n) a 2 , · · · , a n are taken O(1) and a 1 → 0. In [6] one takes a 1 , · · · , a n−1 =O(1) and a n → ∞. The reason of the different choice relies upon tecnicalities related to the evaluation of the "vertical torsion" (i.e. , the entries of the torsion matrix in (2.17) with indices from n + 1 to 2n) in the spatial case. The asymptotics in [10] and [11] does not allow (as in [6]) to evaluate at each step the new torsion simply picking the dominant terms, because of increasing errors (of O(1)): compare the discussion in [11, end of p. 23]. To overcome these technicalities (and to avoid too many computations), Herman introduc! es a modification of the Hamiltonian and a new fictictious small parameter δ, also used in [10]. Notice that, since Herman computes the asymptotics using Poincaré variables, by the presence of the 0-eigenvalue ς n , he could not use the limit a n → ∞, being such limit singular (not continuous) for the matrices ρ v in (2.10). 13 The evaluation of the planar three-body torsion (5.13) is due to Arnold. Compare [2, p.138, Eq. (3.4.31)], noticing that in [2] the second order Birkhoff invariants are defined as one half thē τ ij 's and that a 4 2 should be a 7 2 . Compare also with [11, beginning of p. 21], (where a factor a 3 2 at denominator of each entry is missing).
14 Compare (5.14) and (5.15) with the inductive formulae obtained in the other asymptotics in [11, end of p. 21].
Therefore, the following corollary follows at once.
Proof For ease of computations, we shall consider the functions e i := e 2 i and i j := 1 − cos 2ι j (6.28) and we shall check that, for anyc > 0, one has which implies, clearly, (6.27). The proof of (6.29) comes from the relation between e i , i j and the variables (Λ,ľ,w); in particular, on how e i and i j are related to the stable actions Λ 1 , · · · , Λ n ,ř 1 , · · · ,ř 2n−2 .

A BNFs and symmetries
In this appendix we analyze the properties of Birkhoff-normalizationsφ used in (4.10) for, respectively, partial and total reduction in case of symmetries.

B Domains of Poincaré and RPS variables
In this appendix, for completeness, we describe analytically the global domains M 6n max,p , M 6n max,rps .
• The domain M 6n max,p is the subset of (Λ, λ, z) ∈ R n + ×T n ×R 4n where their respective action variables satisfy where the action variables Γ i , Θ i are regarded as functions of the Poincaré variables in (3.7) i.e. , max,rps is the subset of (Λ, λ, z) ∈ R n + ×T n ×R 4n where 26 the action variables satisfy Here, Γ i , Ψ i are regarded as functions of the rps-variables as in (3.8), i.e. , Notice in particular that the only inequality in (B.2) involving (p n , q n ) is the third one.

D Properly-degenerate averaging theory
In this Appendix we shall prove a result in averaging theory, which is needed in the proof of Theorem 6. where the average P av := T n 1 P (I, ϕ, p, q; µ) dϕ (2π) n 1 has an elliptic equilibrium in p = q = 0 for all I ∈ V . Assume that the map I → ∂ 2 H 0 (I; µ) is a diffemorphism of V ; that the first order Birkhoff invariants Ω of P av do not satisfy resonances on V up to the order 2s. Let τ > n − 1.
There exist positive numbers c , c 0 such that, for all 0 < a < 1 4(τ +1) one can find a number 0 < < 1 such that for all where N and P , f are as above and |Q | ≤ 1.